Incompleteness of Effective Double Frames
Andrzej Wiśniewski,1 Jerzy Pogonowski2
By a double frame we understand any ordered triple (Φ, Ψ, R) such that Φ and Ψ are distinct non-empty sets and R ⊆ Ψ × Φ is a relation with the domain Ψ. For any x ∈ Ψ let: R→x = {z ∈ Φ : xRz}. If Φ and Ψ are subsets of ω (= the set of all natural numbers), then (Φ, Ψ, R) is called a numerical double frame.
We say that a double frame (Φ, Ψ, R) is:
• globally infinite, if Φ and Ψ are countably infinite sets;
• locally infinite, if each set R→x is infinite, for all x ∈ Ψ;
• deeply infinite, it it is both globally and locally infinite.
We say that a numerical double frame is effective, if Φ is recursive, Ψ is r.e. (= recursively enumerable) and R is an r.e. relation. In effective double frames each set R→x is r.e., for all x ∈ Ψ. A numerical double frame (Φ, Ψ, R) is perfect, if Φ, Ψ, and R are recursive.
Let ∆Φ be the family of all infinite recursive subsets of a countably infinite recursive set Φ. We say that a globally infinite numerical frame (Φ, Ψ, R) is recursively complete, if:
∆Φ ⊆ {X : X = R→x for some x ∈ Ψ}.
The Recursive Jump Theorem. For any deeply infinite effective double frame (Ψ, Φ, R) there exists an infinite family {Bj : j ∈ ω} of infinite recursive subsets of Φ such that each Bj is different from any R→x, for all x ∈ Ψ.
The Incompleteness Theorem For Effective Double Frames.
No deeply infinite effective double frame is recursively complete.
Corollary. No perfect deeply infinite double frame is recursively complete.
1Chair of Logic and Cognitive Science, Adam Mickiewicz University, Poland.
E-mail: Andrzej.Wisniewski@amu.edu.pl
2Department of Applied Logic, Adam Mickiewicz University, Poland. E-mail:
pogon@amu.edu.pl
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